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Pierre-Francois Loos 2019-11-27 21:26:34 +01:00
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Manuscript/FarDFT.tex
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@ -217,11 +217,7 @@ In the KS formulation of eDFT, the universal ensemble functional (the weight-dep
\begin{equation}
\F{}{\bw}[\n{}{}] = \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}],
\end{equation}
where
\begin{equation}
\Ts{\bw}[\n{}{}] =
\end{equation}
and
where $\Ts{\bw}[\n{}{}]$ is the noninteracting ensemble kinetic energy functional and
\begin{equation}
\label{eq:exc_def}
\begin{split}
@ -232,7 +228,7 @@ and
+ \int \e{\xc}{\bw}[\n{}{}(\br{})] \n{}{}(\br{}) d\br{}.
\end{split}
\end{equation}
are the noninteracting ensemble kinetic energy functional and ensemble Hartree-exchange-correlation (Hxc) functional, respectively.
is the ensemble Hartree-exchange-correlation (Hxc) functional.
Note that the weight-independent Hartree functional $\E{\Ha}{}[\n{}{}]$ causes the infamous ghost-interaction error (GIC) \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} in eDFT, which is supposed to be cancelled by the weight-dependent xc functional $\E{\xc}{\bw}[\n{}{}]$.
From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain \cite{Gross_1988b,Deur_2019}
@ -243,10 +239,12 @@ From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain
= \Eps{I}{\bw} - \Eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}(\br{})},
\end{equation}
where
\begin{equation}
\n{}{\bw}(\br{}) = \sum_{I=0}^{\Nens-1} \ew{I} \n{}{(I)}(\br{})
\end{equation}
is the ensemble density,
\begin{align}
\n{}{\bw}(\br{}) & = \sum_{I=0}^{\Nens-1} \ew{I} \n{}{(I)}(\br{}),
&
\n{}{(I)}(\br{}) & = \sum_{p}^{\Norb} \ON{p}{(I)} [\MO{p}{\bw}(\br{})]^2
\end{align}
are the ensemble and individual one-electron densities, respectively,
\begin{equation}
\label{eq:KS-energy}
\Eps{I}{\bw} = \sum_{p}^{\Norb} \ON{p}{(I)} \eps{p}{\bw}
@ -715,13 +713,13 @@ This would be, for example, the case with the exact xc functional.
Extracting excitation energies from Eqs.~\eqref{eq:bEwHF}, \eqref{eq:bEwLDA} and \eqref{eq:bEweLDA} is more tricky.
To do so, we will employ Eq.~\eqref{eq:dEdw}.
The two first terms are
The two first terms are simply
\begin{align}
\Eps{0}{\ew{}} & = 2(1-\ew{}) \eps{1}{\ew{}},
\Eps{0}{\ew{}} & = 2 \eps{1}{\ew{}},
&
\Eps{1}{\ew{}} & = 2 \ew{} \eps{2}{\ew{}},
\Eps{1}{\ew{}} & = 2 \eps{2}{\ew{}},
\end{align}
where the HF, LDA and eLDA weight-dependent orbital energies are
and the HF, LDA and eLDA weight-dependent orbital energies are
\begin{subequations}
\begin{align}
\eps{1}{\ew{},\HF}
@ -738,14 +736,14 @@ where the HF, LDA and eLDA weight-dependent orbital energies are
\eps{1}{\ew{},\LDA}
& = \eHc{1} + 2(1-\ew{}) \eJ{11} + 2\ew{} \eJ{12}
\\
& + \frac{1}{2} \int \left. \fdv{\E{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{(0)}(\br{}) d\br{},
& + \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} + \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] } \n{}{(0)}(\br{}) d\br{},
\end{split}
\\
\begin{split}
\eps{2}{\ew{},\LDA}
& = \eHc{2} + 2(1-\ew{}) \eJ{12} + 2 \ew{} \eJ{22}
\\
& + \frac{1}{2} \int \left. \fdv{\E{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{(1)}(\br{}) d\br{},
& + \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} + \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] } \n{}{(1)}(\br{}) d\br{},
\end{split}
\end{align}
\end{subequations}
@ -756,13 +754,13 @@ where the HF, LDA and eLDA weight-dependent orbital energies are
\eps{1}{\ew{},\eLDA}
& = \eHc{1} + (1-\ew{})(2\eJ{11} - \eK{11}) + \ew{}(2\eJ{12} - \eK{12})
\\
& + \frac{1}{2} \int \left. \fdv{\bE{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{(0)}(\br{}) d\br{},
& + \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} + \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] } \n{}{(0)}(\br{}) d\br{},
\end{split}
\\
\begin{split}
\eps{2}{\ew{},\eLDA} & = \eHc{2} + (1-\ew{})(2\eJ{12} - \eK{12}) + \ew{}(2\eJ{22} - \eK{22})
\\
& + \frac{1}{2} \int \left. \fdv{\bE{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{(1)}(\br{}) d\br{},
& + \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} + \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] } \n{}{(1)}(\br{}) d\br{},
\end{split}
\end{align}
\end{subequations}